。
She filed a damages suit against him.
Literal
She [topic-は] him [object-を] taking-as-opponent, damages [genitive-の] lawsuit [object-を] raised.
相手取る (あいてどる) is a legal/formal verb meaning 'to take (someone) as one's opponent' — used almost exclusively in the context of filing suits or pressing charges. 損害賠償 ('compensation for damages') and 訴訟を起こす ('to file a lawsuit') are the standard Japanese legal vocabulary for civil litigation. The te-form 相手取って sets up the manner/circumstance for the main clause: 'taking him as opponent, [she] filed.' The sentence is heavy on 漢語 (Sino-Japanese vocabulary), which is what gives legal Japanese its dry, formal flavor.